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Supplement Part I

Supplement Part I. Map Coloring. B. B. A. A. D. D. C. C. Different Renditions of the Same Graph. Watch the graph change shape! . B. A. D. C. Planar Graphs. A. B. D. C. A graph is PLANAR if it can be drawn without crossing edges.

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Supplement Part I

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  1. Supplement Part I Map Coloring

  2. B B A A D D C C Different Renditions of the Same Graph Watch the graph change shape!

  3. B A D C Planar Graphs A B D C A graph is PLANAR if it can be drawn without crossing edges. The graph the left is planar because we can redraw it (on right) without any crossing edges.

  4. Graphs That Are Not Planar The following graphs cannot be drawn without crossing edges. Try and see why.

  5. The 4 Color Theorem Graphs obtained from MAPS are planar. The 4 color theorem is a new result (c.1976 by Appel and Haken) Any map can be colored with 4 colors or fewer.

  6. Map coloring • Color this map using at most 4 colors, so that no regions that share a border are the same color

  7. Strategy • Create a GRAPH from the map • Make a VERTEX for each country H B L F P Sw I S

  8. Strategy • Create a GRAPH from the map 2. Connect 2 countries if they share a border H B L F P Sw I S

  9. Strategy • Create a GRAPH from the map 3. Write the DEGREE of each vertex on the graph H 1 B 3 5 2 L F 1 2 P Sw 2 2 I S

  10. Strategy • Coloring the Graph 4. Choose 4 colors: purple H 1 green B 3 red 5 2 L F 1 2 P black Sw 2 2 I S

  11. Strategy • Coloring the Graph 5. Starting with the highest degree Vertex (F), color it with the top color purple H 1 green B 3 red 5 2 L F 1 2 P black Sw 2 2 I S

  12. Strategy • Coloring the Graph 6. Color the next highest degree vertex that is connected to F. This is B. Color it green. purple H 1 green B 3 red 5 2 L F 1 2 P black Sw 2 2 I S

  13. Strategy • Coloring the Graph 7. We have a choice now, since there are several vertices of degree 2 connected to F. It does not matter which of these is colored next, as long as we follow the coloring rules. purple H 1 green B 3 red 5 2 L F 1 2 P black Sw 2 2 I S

  14. Strategy • Coloring the Graph 8. Sw is colored green, as we always use the highest colorpossible. purple H 1 green B 3 red 5 2 L F 1 2 P black Sw 2 2 I S

  15. Strategy • Coloring the Graph 9. I is colored red, as we always use the highest colorpossible. purple H 1 green B 3 red 5 2 L F 1 2 P black Sw 2 2 I S

  16. Strategy • Coloring the Graph 10. S is then colored green. purple H 1 green B 3 red 5 2 L F 1 2 P black Sw 2 2 I S

  17. Strategy • Coloring the Graph 11. The last (lowest degree) vertices are then colored. We use purple, since we always use the top color available purple H 1 green B 3 red 5 2 L F 1 2 P black Sw 2 2 I S

  18. Coloring the map • The graph shows the colors to use on the original map.

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